CC BY
0Ke et al. Journal of Uncertainty Analysis and
Applications (2017) 5:5
DOI 10.1186/s40467-017-0059-2
Journal of Uncertainty Analysis and Applications
RESEARCH
Open Access
Pricing Decision in a Two-Echelon Supply Chain with Competing Retailers Under Uncertain Environment
CrossMark
Hua Ke1, Yong Wu1, Hu Huang1* € and Zhiyi Chen2
Correspondence: huanghu0213@163.com 1 School of Economics and Management,Tongji University, Siping Road, Shanghai 200092, China
Full list of author information is available at the end of the article
Abstract
This paper explores a supply chain pricing competition problem in a two-echelon supply chain with one manufacturer and two competing retailers. The manufacturing costs, sales costs, and market bases are all characterized as uncertain variables whose distributions are estimated by experts' experienced data. In consideration of channel members' different market powers, three decentralized game models are employed to explore the equilibrium behaviors in corresponding decision circumstances. How the channel members should choose their most profitable pricing strategies in face of uncertainties is derived from these models. Numerical experiments are conducted to examine the effects of power structures and parameters' uncertain degrees on the pricing decisions of the members. The results show that the existence of dominant powers in the supply chain will increase the sales prices and reduce the profit of the whole supply chain. It is also found that the supply chain members may benefit from higher uncertain degrees of their own costs while the other supply chain members will gain less profits. Additionally, the results demonstrate that the uncertainty of the supply chain will make end consumers pay more. Some other interesting managerial highlights are also obtained in this paper.
Keywords: Two-echelon supply chain, Pricing competition, Competing retailers, Uncertain variable, Stackelberg game
Springer Open
Introduction
In this paper, we explore a pricing decision problem in a decentralized supply chain with competing retailers under uncertainty. In such supply chains, costs and demands may be subject to some inherent indeterministic factors, such as material cost changes, labor costs, technology improvements, and customer incomes. It is necessary to identify distributions of these parameters, which are essential for supply chain managers when they make decisions, e.g., choosing prices, determining production quantities, choosing ordering quantities, and adding investment. Theoretically, we can collect enough samples to estimate distributions of these parameters before we make decisions. However, products, especially high-tech products, e.g., smartphones, PCs, and other digital devices, have been rapidly updated recently. The costs and demands for these new products are usually the lack of historical data, or we can collect enough data, but these data may not be applicable due to the dynamic environment. Practically, domain experts' experience data
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(belief degrees) are often applied to estimate the distributions in such cases. Nevertheless, some surveys have shown that human beings (even the most experienced experts) usually estimate a much wider range of values than the object actually takes. It makes belief degrees behave quite different from frequency, indicating that human belief degrees given by managers and experts should not be treated as random variables or fuzzy variables [1]. Instead, based on normality, duality, subadditivity, and product axioms, uncertainty theory, initiated by Liu [2] and refined by Liu [3], can be introduced to deal with parameters estimated by human belief degrees.
The goal of this paper is to study supply chain pricing problems with uncertainty theory. Our work mainly focuses on equilibrium behaviors of the decentralized supply chain with one common manufacturer (upstream member) and duopoly retailers (downstream members) under uncertainty. A supply chain with competing retailers is not uncommon in the real industry world. For instance, smartphone producers distribute their products through both traditional retailers and online retailers. How should the supply chain managers choose the most profitable price policies with experts' estimations? What effects might the uncertain degrees of the parameters have on the supply chain members' pricing decisions and expected profits? Additionally, we assume that the manufacturer determines the wholesale prices and the two retailers compete for end consumers by choosing their own sales prices. The decision sequence is mainly decided by the power structures of the supply chain. In some supply chains, the manufacturers (like Intel and Microsoft) often play a more powerful role with regard to the other channel members. In some other chains, the retailers (like Warmart and Carrefour) who are much bigger than most manufacturers often hold the dominant power. While in another type of supply chain, no absolute dominance exists between different members and each member holds the same power. How would these various power structures affect the performance of the supply chain?
In order to answer these questions, three decentralized pricing models based on uncertainty theory and game theory are built. Then, the corresponding closed-form equilibrium solutions are derived from these models. Afterwards, numerical experiments are conducted to analyze the equilibrium behaviors of the supply chain under different power structures and uncertain settings. It is found that the manufacturer can benefit from a higher uncertain degree of the manufacturing cost, meanwhile, the retailers will gain less profits. Similarly, when the uncertain degree of the sales costs increases, the retailers can make more profits while the manufacture will gain less. Additionally, the uncertainty of the supply chain will make end consumers pay more. Some other interesting managerial highlights are also obtained in this paper.
The remainder of this paper is organized as follows: some related literatures are presented in the "Literature Review" section. Some basic concepts and properties with respect to uncertain variable are reviewed in the "Preliminaries" section. Some useful notations and necessary assumptions are discussed in the "Problem Description" section. Three models are employed to derive the equilibria under three possible scenarios in the "Models and Solution Approaches" section. In the "Numerical Experiments" section, numerical experiments are applied to examine the equilibrium behaviors of the supply chain members. Some conclusions and possible extensions are discussed in the "Conclusions" section.
Literature Review
By now, the pricing competition problem in decentralized supply chains, initiated in 1980s [4, 5], has been well studied by both scholars and practitioners. Most of these researches on the pricing competition problem focus on the following four structures: monopoly common retailer structure [6-8], chain-to-chain structure [4,9,10], dual channel structure [11-14], and monopoly common manufacturer structure, which is studied in this paper. The research on the monopoly common manufacturer structure was initiated by Ingene and Parry [15], considering a coordination problem in a supply chain where a manufacturer sells its products through competing retailers. Ingene and Parry [16] demonstrated that the manufacturer can generally obtain more profits by setting a unique two-part tariff wholesale pricing policy. For pricing competition, Yang et al.[17] considered a pricing problem in a two-echelon supply chain with a manufacturer who supplies a single product to two competing retailers. Assuming that the manufacturer acts as a Stackelberg leader, they explored the effects of the retailers' Bertrand and Collusion behaviors on the performance of the supply chain. Wu et al.[18] investigated the pricing decision in this monopoly common manufacturer channel structure and built six noncooperative models where the two retailers play Stackelberg or Bertrand games under three possible power structures. Besides, Huang et al.[19] considered a pricing competition and cooperation problem in a supply chain with one common manufacturer and duopoly retailers and explored the conditions under which the retailers can benefit from their collusion behaviors.
The work above typically focused on deterministic demands and costs. In fact, the real world exists many indeterminate factors which cannot be ignored when making pricing decisions. Thus, one important tendency is to explore the pricing competition problem under environments with indeterminate factors, namely randomness, fuzziness, or uncertainty. For instance, Bernstein and Federgruen [20] investigated the equilibrium behaviors of decentralized supply chains with competing retailers under random demand and designed some contractual arrangements between the parties that allow the decentralized chain to perform like a centralized one. Xiao and Yang [21] studied a price and service competition of two supply chains with risk-averse retailers under stochastic demand. They also analyzed the impacts of the retailer's risk sensitivity on the manufacturers' equilibrium strategies. Wu et al.[9] considered a joint pricing and quantity competition between two separate supply chains in random environment and explored the effects of randomness on the equilibrium behaviors of the supply chain members. Shi et al.[22] utilized a game-theory-based framework to formulate the power in a supply chain and examined how power structure and demand indeterminacy affect the supply chain members' performances. Mahmoodi and Eshghi [23] studied this pricing problem in duopoly supply chains with stochastic demand and explored the effects of competition and demand indeterminacy intensity on the equilibria of the structures by a numerical experiment. Li et al.[24] studied the pricing and remanufacturing decision problem with random yield and demand.
More recently, fuzzy set theory has been introduced to the pricing decision problems. Zhou et al.[25] considered the pricing decision problem in a supply chain which consists of a manufacturer and a retailer in fuzzy environment. Zhao et al.[26] and Sang [27] studied the pricing problem on two substitutable products in a supply chain with different structures in which the manufacturing costs and consumer demands were described by
fuzziness. Specially, Liu and Xu [28] and Ke et al.[29] studied the pricing problem in a fuzzy supply chain consisting of one manufacturer and two competitive retailers.
To the best of our knowledge, there are little researches on the pricing decision problem with competing retailers under uncertain settings. Differing from the literatures above, this paper addresses the pricing equilibria under circumstances with only belief degrees being available, which can be described by uncertainty theory. By far, the new theory has been successfully applied to deal with many uncertain decision-making problems, e.g., option pricing [30, 31], facility location [32], portfolio selection [33], inventory problem [34], project scheduling problem [35-37], and production control problem [38]. Recently, Huang and Ke [8] applied uncertainty theory to a pricing decision problem in a supply chain with one common retailer. Besides, Chen et al.[39] studied an effort decision problem in a supply chain with one manufacturer and one retailer under uncertain information. However, both of them did not consider the pricing problem in a supply chain with competing retailers.
This paper extends the above literatures on the pricing problems with competing retailers [17-19, 28], by employing uncertainty theory to depict the costs and demands estimated by experts' experience data in practice. Three uncertain game models are built to explore the equilibrium behaviors of the decentralized supply chain under different power structures. Meanwhile, we consider the retailers' sales costs which are often ignored in most literatures. Then, we explore the effects of uncertain degrees of the manufacturing costs and sales costs on the equilibrium prices determined by the supply chain managers. Some interesting managerial highlights are gained from some numerical experiments conducted in this paper.
Preliminaries
Let r be a nonempty set and L a a-algebra over r. Each element A in L is called an event.
Definition 1 ([2]) The set function M is called an uncertain measure if it satisfies:
Axiom 1 (Normality axiom) M{r} = 1.
Axiom 2 (Duality axiom) M{A} + M{Ac} = 1 for any event A.
Axiom 3 (Subadditivity axiom) For every countable sequence of events {A,}, i = 1,2, • ••, we have
Besides, the product uncertain measure on the product a-algebra L was defined by Liu [40] as follows:
Axiom 4 (Product axiom) Let (r^, Lk, Mk) be uncertainty spaces for k = 1,2, ••• The product uncertain measure M is an uncertain measure satisfying
M M{A,}.
M Mk{Ak}
where Ak are arbitrarily chosen events from Lkfor k = 1,2, •••, respectively.
Definition 2 ([2]) An uncertain variable is a measurable function f from an uncertainty space (r, t, M) to the set of real numbers, i.e., for anyBorel setB of real numbers, the set
{f e B} = {y e r \f(y) e B}
is an event.
Definition 3 ([40]) The uncertain variables f1, f2, ••• , fn are said to be independent if
for any Borel sets B1, B2, ••• , Bn of real numbers.
Sometimes, we should know uncertainty distribution to model real-life uncertain optimization problems.
Definition 4 ([2]) The uncertainty distribution $ of an uncertain variable f is defined by
$(x) = M{f < x} for any real number x.
An uncertainty distribution $ is referred to be regular if its inverse function $-1(a) exists and is unique for each a e [0,1].
Lemma 1 ([3]) Let f1, f2, ••• , fn be independent uncertain variables with regular uncertainty distributions $1, $2, ••• , $n, respectively. If the function f (x1, x2, • • • , xn) is strictly increasing with respect to x1, x2, ••• , xm and strictly decreasing with respect to xm+1, xm+2, • • • , xn, then
Definition 5 ([2]) Let f be an uncertain variable. The expected value of f is defined by
provided that at least one of the above two integrals is finite.
Lemma 2 ([3]) Let f be an uncertain variable with uncertainty distribution $. If the expected value exists, then
Lemma 3 ([3]) Let f be an uncertain variable with regular uncertainty distribution $. If the expected value exists, then
$-\a) = f ($-1(a), ••• , $m1(a), $^(1 - a), ••• , $-1(1 - a)).
Lemma 4 ([41]) Let f1, f2, ••• , fn be independent uncertain variables with regular uncertainty distributions $1, $2, ••• , $n, respectively. A function f (x1, x2, • • • , xn) is strictly increasing with respect to x1, x2, ••• , xm and strictly decreasing with respect to xm+1, xm+2, ••• , xn. Then, the expected value of f = f (f1, f2, ••• , fn) is
provided that the expected value E[ f ] exists.
Example 1 Let f and n be two positive independent uncertain variables with regular uncertainty distributions $ and respectively. Then we have
Problem Description
Consider a supply chain with one common manufacturer and two competing retailers: the manufacturer supplies one common product to the two retailers, and in turn, the two retailers compete to sell the product to end consumers in the same market.
Notations and Assumptions
It is assumed that the manufacturer determines the wholesale prices (w1, w2) to the two retailers and then the two retailers choose their own markup policies r1 and r2, respectively. Note that the sales prices can be decided (pi = wi + ri, i = 1,2) when the corresponding wholesale prices and retailer markup policies are determined. Consistent with the extant literatures [6,17-19,42], we suppose that each retailer faces the following linear demand function:
qi = di - fip, + yp3-i, i = 1,2
where di represents the potential market size of retailer i, fi is the self-price elasticity coefficient, y is the cross-price elasticity coefficient, and pi is the sales price of retailer i. Note that y / fi denotes the substitutability of the two retailers.
Owing to the complicated and changeable environment, the manufacturing cost c, sales costs (s1, s2), market sizes (d1, d2), and price elasticity coefficients fi, Y are usually unknown to the supply chain managers. Thus, belief degrees, given by experienced experts, are often employed to estimate these parameters due to the lack of historical data. To sum up, we give the notations shown in Table 1.
Table 1 Notations
c : Unit manufacturing cost, an uncertain variable
si : Unit sales cost of retailer i, an uncertain variable
wi: Unit wholesale price to retailer i, the manufacturer's decision variable
ri: Unit markup price of retailer i, the retailer's decision variable
pi: Unit retail price of retailer i, where pi = wi + ri
di : The market base of retailer i, an uncertain variable
nm: The profit of the manufacturer: nm =
E2=1 (Wi - c)(d, - fipi + Y P3-i)
nrt: The profit of retailer i: nrt = (ri — si)(di — fipi + Y p3-i)
Note that some basic concepts and theorems of uncertainty theory are provided in the "Preliminaries" section. Interested readers can consult Liu [1] for more details of the uncertainty theory. In order to attain closed-form solutions, some necessary assumptions are made as follows:
• As each retailer's demand should be more sensitive to changes of its own price than to changes of the other retailer, it is assumed that the elasticity coefficients 3 and y satisfy: E[¡] > E[ y] > 0.
• All the uncertain coefficients are assumed nonnegative and mutually independent.
• All the supply chain members have access to the same information on the demands and the costs.
• All the supply chain members are risk neutral.
• All the pricing activities are assumed to be completed in a cycle.
• The wholesale prices and markups should exceed the costs and the retailers' demands are always positive:
M{wi - c < 0} = 0, M{ri - si < 0} = 0,
- - (3)
M{di - ¡Pi + YP3-i < 0} = 0, i = 1,2.
Crisp Forms of Objective Functions
In order to derive the equilibrium solutions, we transform uncertain objective functions to equivalent crisp forms first. Let c, Si, di, p, and y be positive independent uncertain variables with regular uncertainty distributions $c, , , ^p, and , i = 1,2.
As M {wi - C < 0} = 0, M {ri - si < 0} = 0, M{di - ppi + y p3-i < 0} = 0, i = 1,2, one can find that nm is monotone increasing function of di, y and monotone decreasing with C, p, and Si. Thus, referring to Lemma 4, the expected object function of the manufacturer can be attained as follows:
|^(wi - Ci) (di - p (ri + wi) + y (r3-i + W3-i))
^ Jo [W - - *)} (^d1(a) - V(1 - ") (ri + wi)
+$-1(a) (r3-i + W3-i))
Define
E[a1-aba] = f1 $-1(1 - a)$-1 (a)da,
E[a1-ab1-a] = i $-1(1 - a)$-1(1 - a)da. J0
where and are the inverse distribution functions of a and b, respectively.
Then, the crisp form of nm can be attained as follows:
nm [ (Wi - (1 - a))(*- (a) - V (1 - a) (n + Wi)
+ $-1(a)(rs-i + W3-'))]da 2
= £{-E[j8] w2 + E[ y] W3-iWi + (-E[j8] n + E[ Y] k-i + E[¿¿] +E[ c1-afi1-a] )w i=1
- E[ c1-ada] +E[ c1-afi1-a] ri - E[ c1-aYa] (T3-i + W3-i)}.
Similarly, the crisp forms of the expected profit functions of the retailers can also be attained as follows:
nn = - E[ p] r2 + E[ Y] r3-iri + (-E[ p] wt + E[ Y] W3- + E[ dt] +E[s1-ap1-a] )n - E[s1-ada] +E[s1-afi1-a] Wi - E[s1-aYa] (r3-i + W3-i), i = 1,2.
Models and Solution Approaches
In this section, starting with the Manufacturer Stackelberg (MS) model, we present the general formulations and solutions to three types of supply chain pricing models with different power structures.
MS Model
In the first case, we assume that the manufacturer plays a dominant role and announces its wholesale prices (w1, w2) to maximize its own profit allowing for the retailers' reaction functions. Observing the wholesale prices, the two retailers noncooperatively choose their own markup pricing schemes ri to maximize their own profits conditional on the other retailer's decision. Then, the retail prices are decided as well as the ordering quantities. Thus, a Stackelberg-Nash game can be applied as follows:
max nm = E E[ (wi - c)(di - fi(rf + Wi) + Y(r3^-i + W3-i))]
W1,W2 i=1
subject to:
M{wi - c < 0} = 0, i = 1,2
where (rjf, r|) solves problems:
max nn = E[ (n - s\)(d1 - ^(n + W1) + Y(r2 + W2))] (8)
max nr2 = E[ (r2 - s"2)(d2 - fir + W2) + Y(r1 + W1))]
subject to: M{ri - Ci < 0} = 0,
M{di - ^(ri + Wi) + Y (r3-i + W3-i) < 0} = 0, i = 1,2.
To solve this Stackelberg-Nash game model, opposite to the decision sequence, we should first derive the Nash equilibrium at the lower level for the given wholesale prices W1 and w2 specified by the manufacturer in advance.
Proposition 1 Given the wholesale prices (wi, w2) the Nash equilibrium in the lower level can be obtained as follows:
- 2E [ ¡3]2 +E[ y ]2 , E[j8] E[ y ]
r1(w1, w2) =---w1 +----w2
u 1 2; 4E[/3]2 -E[ Y]2 1 4E[/3]2 -E[ y]2 2
, 2E[j8] (E[ d1] +E[ ¡31-as1-a]) + E[ y] (E[ d2] +E[33~1-as1-a])
+--1—--2-,
4E[ ¡3]2 -E[ y]2
-2E[/3]2 +E[ y ]2 , E[j8] E[ y ]
r2(w1, w2) =---w2 +----w1
^ 1 ^ 4£[/3]2 -E[ Y]2 2 4£[/3]2 -E[ y]2 1
+ 2E[ ¡3] (E[ d2] +E[ ¡i1-as2-a]) + E[ y] (E[ (dj] +E[;3~1-as1-a])
4E[/3]2 -E[ y]2 .
Proof Referring to the two retailers' expected objective functions with the given wholesale prices, we can get
d2nn r ~ d2nri r~ / x
—= -2E[3S] < 0, —42 = -2E[3S] < 0, (10)
dr2 dr2
with the assumption E[ ¡J] > E[ y] > 0. Hence, nr1 and nr2 are concave in r1 and r2, respectively. Setting the first-order derivatives equaling zero, we have
dnr1 =- 2E[ ¡3] n + E[ y] r2 - E[ ¡3] w1 + E[ y] w2 + E[ d1] +E[s1-a¡S1-a] = 0,
r2 = - 2E[ 3e] r2 + E[ Y] r1 - E[ ¡3] W2 + E[ y] w + E[ d2] +E[S1-apC 1-a] = 0.
The followers' optimal responses to (w1, w2) can be easily obtained by solving the above two equations as follows:
-2E[;3]2 +E[ Y]2 , E[fl E[ Y]
rT(w1, w2) =---w1 +----w2
1V 1 2; 4E[¡S]2 -E[ y]2 1 4E[/3]2 -E[ y]2 2
, 2E[p3] (E[ d1] +E[ ¡i1-a S1-a ] ) + E[ y ] (E[ d2] +E[~p1-a s\-a ] )
+--1—--2-,
4E[ ¡S]2 -E[ y]2
-2E[;3]2 +E[ Y]2 , E[fl E[ Y]
r9(w1, w2) =---w2 +----W1
1 ^ 4E[¡S]2 -E[ Y]2 2 4E[¡S]2 -E[ y]2 1
, 2E[ 0] (E[ d2] +E[ ft1-"S1-a] ) + E[ y] (E[ &] +E[/31-aS1-a] ) +--2—=-1-.
4E[/3]2 -E[ y]2
Thus, Proposition 1 is proved. □
The manufacturer, given the optimal responses of the two retailers, will make decisions to maximize its expected profit.
Proposition 2 Given the optimal reSponSeS of the two retailerS, we can obtain the following:
(1) nm Is jointly concave with reSpect to w1 and w2;
(2) The optimal decisions of the manufacturer are as follows:
„ A(E[ di] —E[/3] Si + E[ y] S2 + C) + B(E[ d2] -E[ /3] S2 + E[ Y] Si + C)
Wi = -.
Wi 2(A2 - B2) '
A(E[ d2] -E[j8] S2 + E[ Y] Si + C) + B(E[ di] —E[fi] Si + E[ Y] S2 + C)
2(A2 - B2)
A = 2E[fi]3 —E[flE[ Y]2 B = E[fi]2E[ Y]
4£[/S]2 —E[ Y ]2 4E[/3]2 —E[ Y ]2
(2E[ fi]2 +E[ fl E[ Y] )(E[ ~ci-api-a] —E[ ci-aYa] ) 4£[fi]2 —E[ Y ]2 .
c 2E[fl (E[ di] +E[fi1-as1-a]) + E[ Y] (E[ d3-i] +E[fi1-as1-a]) ._
Si — ^ , i — 1, 2.
4£[/S]2 -E[ Y]2
Proof Substituting the two retailers' reaction functions Eqs'. (9) into the manufacturer's profit function, and we obtain
nm = ^E[ (Wi — c)(di — fi(r* + Wi) + Y(r*i + Wi))]
Wr -w 5, 2E[fi]2 E[fl E[ Y ]
= > E[ (w' - c)(di - p(---w +----w3—i + S')
[V i PV4£[/i]2 -E[ Y]2 " 4£[/3]2 -E[ Y]2
, 2E[fi]2 , E[flE[Y] _ ...
+ Y(-~-W3- i +----Wi + S3-i))]
M4£[/i]2 -E[ Y]2 3 " 4E[fC]2 -E[ Y]2 " 3 ]
2 (14)
= ^{E[ di] Wi - AW"2 + £W3-iWi - E[ fi] SiWi + E[ Y] S3-iWi
- E[ c1-ada] +E[ c1-afi1-a] Si - E[ c1-aYa] S3-i 2E[ c1-aC1-a] E[fi]2 -E[fi] E[ Y] E[ c1-aYa]
+----W'
4E[/3]2 -E[ Y]2 E[j8]E[ Y]E[c1-aC1-a] -2E[fi]2E[c1-aYa]
+----W3- i}.
4E[/3]2 -E[ Y]2 3
Referring to Eq. (14), we can obtain the Hessian matrix with the corresponding second-order derivatives of the objective function
ttm as follows:
d ^ftm d2^m
d wi 5WI5W2
d 2nm d2nm
d W2 d Wi 3w2
—2A 2B 2B — 2A
With assumption E[/3] > E[ Y] > 0. we can obtain
2E[fi]3 — E[ 3] E[ Y]2 E[fi]3 E[fi]2 E[ Y] _
A =--- > --- > ---= B.
4£[/S]2 —E[ Y ]2 4E[/3]2 —E[ Y ]2 4£[/S]2 —E[ Y ]2
It can be easily seen that H is negative definite and nm is jointly concave with respect to w1 and w2. Therefore, differentiating nm by (w1, w2) and equating the expressions to zero, we have
dnm(w1, w2) d w1
dnm(w1, w2) d w2
= -2Aw 1 + 2Bw2 + E[ d1] -E[ ¡3] 51 + E[ y] 52 + C = 0, = -2Aw2 + 2Bw1 + E[ d2] -E[ ¡3] 52 + E[ y] 51 + C = 0,
and the leader's equilibrium prices (w1, w2) can be easily obtained by solving the above two equations
A(E[ d1] -E[ ¡3] 51 + E[ y] 52 + C) + B(E[ d2] -E[ ¡3] 52 + E[ y] 51 + C)
2 (A2 - B2) '
A(E[d2] -E[¡3 52 + E[ y] 51 + C) + B(E[dj -E[ p] 51 + E[ y] 52 + C)
2 (A2 - B2)
Thus, Proposition 2 is proved.
Based on Propositions 1 and 2, we can get the equilibrium prices of the two retailers as follows:
^-2E 3 2 + E [y wT+ E 3 E [y ] wT + 2E 3 (e ch + E 3~S1 )
- e [9 ]2
E [9] (E
- E [9]2
^-2E 3 2 + E [ywT+ E 3 E [Y] wT + 2E 3 (e ¿2 + E ¡¡S2 )
- e [9 ]2
E [9] (E
- E [9]2
RS Model
The second possible structure is that the manufacturer distributes its products through two supper retailers which are much bigger than the manufacturer. We assume that the
two retailers can dominate the sales prices by choosing their markup policies firstly. Then, a Nash-Stackberg model can be applied as follows:
niax nn = E[ (ri - ~s\)(d\ - p(ri + w\) + y(r2 + w£))]
max nr2 = E[ (r2 - ~S2)(d,2 - P(r2 + w£) + y(ri + w*))]
subject to: M(r; -Si < 0} = 0, i = 1,2
where (w*, w2) solves problem: (16)
max nm = J2 E[ (wi - c)(di - p(n + w,-) + y(r3-i + w3-i))] wi,w2 i=i
subject to:
M{di - P(n + wi) + Y(r3-i + w3-i) < 0} = 0, M{w,- - S < 0} = 0, i = 1,2.
Similarly, opposite to the decision sequence, we should derive the follower's response to the given retail markups r1 and r2 specified by the two retailers.
Proposition 3 Observing the two retailers' retail markups r1 and r2, the manufacturer's optimal response is:
E[ p] E[ di] +E[ y] E[ d2] + (E[ P] +E[ y] )(E[ C1-ap1-a] -E[ C1-aya]) 1
wi (ri, r2) = -^---ri,
1 2(E[S]2 -E[ y]2) 2 1
E[ p] E[ d2] +E[ y] E[ di] +(E[ p] +E[ y] )(E[ ox-ap1-a] -E[ C1-aya]) 1
w2(r1, r2) = -=---r2.
, 2(E[ p]2 -E[ y]2 ) 2 2
Proof Referring to Eq.(6), the Hessian matrix with the corresponding second-order derivatives of the equivalent objective function nm can be attained:
d nm dwf d w1d w2 d 2nm d 2nm
3w23w1 d w2
-2E[fi] 2E[y] 2E[y] -2E[fi]
Obviously, the Hessian matrix is negative definite with the assumption E[ p] > E[ y ] > 0. Then, the optimal reaction functions to (r1, r2) can be derived from the first-order conditions.
w1 + 2E [y ] w2 + E
- E[C1-aya] = 0,
= - 2E
w2 + 2E [y ] w1 + E
n + E [y ] r2 + E
r2 + E [y ] r1 + E
c1-aS1-a
1-a 21-a
- E [C1-aya] = 0.
By solving Eqs. (18), we can obtain:
E[ p] E[ dj] +E[ y] E[ d2] +(E[ p] +E[ y] )(E[ C1-ap1-a] -E[ C1-aya]) 1
w1 (ri, r2) = -^---ri,
1 2(E[ p]2 -E[ y]2 ) 2 1
E[ fl E[ d2] +E[ y] E[ di] +(E[ p] +E[ y] )(E[C1-a^1-a] -E[C1-aya]) 1
w2(r1, r2) = -^---r2.
1 2(E[ p]2 -E[ y]2 ) 2 2
Thus, Proposition 3 is proved.
Given the manufacturer's response, the two retailers compete to choose their own sales markups.
Proposition 4 The Nash equilibrium between the two retailers is as follows:
2E P E d 1 + E [y ] E d 2 — 2E P + E [y])e c1—a 1—a — E [c1—aYa]
4E P 2 "E [Y]2
2E p E 's\—ap 1—a + E [Y ] E 's12—ap1—a'
4E P 2 t — E [Y]2
2E p E d2 + E [Y ] E d1 — 2E P + E [Y])e c1—a 1—a — E [c1—aYa]
4E P 2 -E [Y]2
2E p E 's2—ap 1—a + E [Y ] e 's\—ap1—a'
4E p 2 t — E [Y]2
Proof Substituting Eqs. (17) into the two retailers' profit functions, we obtain:
nr, =--E
r2 + 1E [y ]r2ri + 2 (E
s\—ap1—a
ri - e
nr2 = — 2E +E s:
si-a dax 1
rf + 1E [?]rir2 + 1 (E
,1— a q 1—a r ^1—aja
di + E d2
~1—a a 1—a
+ E[c1—aya]
=1—a ¿T 1—a
D1 — E
-1 — a~ a
2 u j 1 2 2 n — E Ts2—a d2 + E s2
~l—ao 1—a
+ E[c1—aya]
1—a s1—a D2 — E s1—aYa |D1 + ■
[fi] E [dj + E [y] E [d3-^ + (e [/8] + E [y]) (E [c1—ap1—1— E [C1—aya])
2 (e [P]2 — E [Y]
i = 1,2.
With the second-order condition = —E[ p] < 0, the first-order conditions can be
shown as follows:
ln d r1 dnr2 d T2
[8] n + 2E [y] r2 + 2 (e d] — E [c1—aè1—a] + E [c1—aYa] + E [s1—^ = 0, [8] r2 + 2E [y] n + ! (E [d2] — E [c1—ap1—a] + E [C1—aYa] + E [s\—ap1—aJ) = 0.
By solving Eq. (23), the optimal decisions of the two retailers can be derived:
+ E [y ]E di - (2E p + E [y] )e c1—ap1—a — E[c1-aya]
;1—a 21-a
+ E [y ] E
— E [Y f
~1—a 21—a
— E [Y]2
+ E [Y ]E d1 — (2E fi + E [Y] )e c1—1—a — E[c1—aYa]
— E [Y ]2
;1—a« 1—
+ E [Y ] E
1—a 1—
— E [Y ]2
Thus, Proposition 4 is proved.
Based on Propositions 3 and 4, we can get the equilibrium prices of the manufacturer as follows:
E [>] E [¿1] + E [Y] E [d2] + (E [>] + E [Y]) (E [c1—ap1—^ — E \c1—aYa]) 1 +
TTT2 T^K 2r1,
2 E -r--,^
[ ë] — E [y]
[p] E [¿2] + E [Y] E [¿1] + (e + E [Y]) (E [c1—ap1—a] — E [C1—aYa])
■[ H] — E [Y ]:
VN Model
In some supply chains, there is no obvious dominance between chain members. In this case, we assume that the three players move simultaneously. Then, a three-player Nash game model can be built as follows:
max nm = V E
£ E (wi — c) (di — p (ri + wi) + Y (T3—i + W3—,}) 1=1 L v /J
max nr, = E
max = E
(r1 — S1) (¿1 — P (T1 + W1) + Y (r2 + W2))
(T2 — S2) (r1 + W1)
subject to:
M {wi — c < 0} = 0, M {ri — si < 0} = 0, M di — P (ri + wi) + Y (r3—i + W3—i) < 0} = 0, i = 1,2.
Proposition 5 In the Nash equilibrium of the three-player game, the equilibrium prices are obtained as follows: l2
¡3 2 + 2E [Y Di + 4E ¡3 E [Y ] D2 + 6E ¡3 (e
si-a31-a
- E [Y f
2E [9] (E
;1-a« i-
- E[ Y ]2
+ 2E [Y]2 ) D2 + 4E [¡îj E [Y] Di + 6E
s2~a ¡31-a
- E [Y]2
2E Y E
;1-® /51-a
- E [Y]2
^12E ¡3 2 - 2E [Y]2^ D1 - 2E ¡3 E [Y] D2 - 3E ¡3 (e
1-^ 1-a
si-ay3
- E [Y]2
E [Y] (E d2 + E s12-a 1-a )
9E 2 t - E [Y]2
^12E ¡3 2 - 2E [Y]2^ D2 - 2E ¡3 E [Y] D1 - 3E ¡3 (e
s12-a 1-a
- E [Y]2
E [Y] (E d1 + E s11-a31-a )
9E 2 t - E [Y]2
Proof Note that nm is jointly concave in (w1, w2) and nn is concave with respect to ru i = 1,2. Set the first-order derivatives equaling zero as follows:
W1 + 2E [Y ]W2 - E ¡3 r1 + E [Y ]r2 + E
- E[c1-aYa] = 0,
W2 + 2E [Y ]W1 - E ¡3 r2 + E [Y ]n + E
c1-a a
c1-a a
- E[c1-aYa] = 0,
dnr1 d r1 dnr2 d r2
n + E [Y ]r2 - E ¡3 W1 + E [Y ]w2 + E r2 + E [Y ]r1 - E ¡3 W2 + E [Y ]W1 + E
+ E + E
s12-a a
s12-a a
= 0. (27)
By solving Eqs.(27), the equilibrium solutions of Eqs.(26) can be attained. □
Similarly, the sales prices, ordering quantities, and expected profits can also be attained.
Numerical Experiments
By the results obtained from the above models, the expressions of each supply chain member's optimal decision and optimal expected profit can be easily attained. Because of the complicated forms of the equilibrium prices and expected profits, it is quite difficult (if possible) to conduct analytical comparisons to obtain general conclusions. Instead, we employ numerical approach to illustrate supply chain members' different behaviors and performances facing various decision environments with different power structures and uncertain degrees.
When dealing with indeterminate quantities (e.g., demands and costs) without enough historical data (samples), experienced experts are usually employed to estimate these quantities in practice and hence uncertainty theory can be applied. Interested readers can consult Liu [1] (Chapter 16: Uncertain Statistics) to get more details on how to collect experts' experimental data and how to estimate empirical distribution of uncertain variable from the experimental data. For simplicity, we just list uncertainty distributions of the uncertain parameters in Table 2.
Additionally, this paper uses E[ y] /E[p] to represent the substitutability of the two retailers. Let fi remain constant and vary y at three levels, we show the data in Table 3.
With Lemma 3, we can attain E[ c] = (9 + 11)/2 = 10, E[Si] = (5 + 7)/2 = 6, E[S2] = (4 + 6)/2 = 5, E[ ¿1] = (2900 + 2 x 3000 + 3300)/4 = 3050, E[ ¿2] = (2800 + 2 x 3000 + 3100)/4 = 2975, etc.
According to Lemma 4, we have
(11(1 - a)) + 9a)(2900 x (1 - 2a) + 2 x 3000a)da
=3043.33,
=496.67.
Table 2 Distributions of uncertain variables
Parameter
Distribution
Expected value
L(9,11) L(5,7) L (4,6)
d1 d2 ß Y
L (80,120) L(40,60)
Z(2900,3000,3300) Z(2800,3000,3100)
2975 100
Table 3 Substitutability of the two retailers
Substitutability Y Expected value E[ Y ] /E[ß]
Low L(15,35) 25 0.25
Medium L(40,60) 50 0.50
High L(70,80) 75 0.75
In the same way, we can get the values of E[c1 1 a], E[c1 aYa], E[S1 afi1 a], E[s1-aYa], etc.
Substituting these values into the equilibrium solutions attained in the "Models and Solution Approaches" section, we can obtain Table 4.
Referring to Table 4, one can find the following:
• Regardless of the leadership, the sales prices and the expected profits of the total system in the MS and RS structures are the same, indicating that no matter who holds the power has no influence on the total supply chain. However, in consideration of the individual firms, the firm acts as a leader will gain more profit than as a follower.
• The dominant power structure will increase the sales prices and lower the profits of the whole supply chain as the sales prices in the VN case are the lowest.
Next, we analyze the effects of the uncertain degrees of the manufacturing costs and sales costs on the optimal pricing decisions under the three possible structures. The uncertain degree of a parameter mainly depends on its own inherent variance and experts' personal knowledge. If more information on a parameter is available, more accurate estimations the experts can make. Then, as a result, the uncertain degree of the parameter will decrease. Of course, if we have enough information (historical data) on these parameters and their distributions can be precisely estimated, then we can apply probability theory rather than uncertainty theory. Note that the forms of the equilibrium solutions of these models under stochastic environment are the same to those under deterministic environment with the assumption of risk neutral members.
By varying the uncertain degrees of these parameters and keeping the other parameters unchanged as shown in Table 2, the changes of optimal prices are shown in Tables 5, 6, and 7.
Referring to Table 5, we obtain the following results:
Table 4 The equilibrium results of the three structures under different substitutability
Sub. Str. w w2 nm r1 p1 nri r2 p2 nr2 nt
0.25 MS 22.2667 22.4667 13,748.63 11.2540 33.5206 3124.97 10.3651 32.8317 3656.1 1 20,529.71
VN 18.9688 19.0918 12,794.36 12.6625 31.6312 4775.00 11.8163 30.9082 5416.60 22,985.96
RS 17.0794 17.1683 8250.51 16.4413 33.5206 5815.78 15.6635 32.8317 6463.42 20,529.71
0.5 MS 32.3167 32.5667 34,302.99 13.4056 45.7222 5956.74 12.5556 45.1222 6613.49 46,873.23
VN 27.9219 28.0648 32,983.47 14.8562 42.7781 8276.47 14.0705 42.1352 9124.66 50384.61
RS 24.9778 25.0778 23,308.72 20.7444 45.7222 11,342.67 20.0444 45.1222 12,221.84 46,873.23
0.75 MS 62.5238 62.8095 111,801.44 16.4291 78.9529 11,661.49 15.6109 78.4204 12,492.06 135,954.99
VN 56.7831 56.9354 110,451.84 17.5481 74.3312 14,067.03 16.8148 73.7503 15,179.93 139,698.79
RS 52.1614 52.2653 89,945.35 26.7915 78.9529 22,399.47 26.1552 78.4204 23,610.17 135,954.99
Table 5 Effects of the manufacturing cost's uncertain degree on the prices and profits
c W1 w2 nm n P1 n P2 n,2 nt
MS 10 32.2167 32.4667 33,278.55 13.4389 45.6556 6005.34 12.5889 45.0556 6663.09 45,946.97
£(9,11) 32.3167 32.5667 34,302.99 13.4056 45.7222 5956.74 12.5556 45.1222 6613.49 46,873.23
L(8,12) 32.4167 32.6667 35,328.77 13.3722 45.7889 5908.37 12.5222 45.1889 6564.12 47,801.27
£(7,13) 32.5167 32.7667 36,355.88 13.3389 45.8556 5860.23 12.4889 45.2556 6514.98 48,731.08
VN 10 27.8019 27.9448 31,947.14 14.8962 42.6981 8346.42 14.1105 42.0552 9196.32 49,489.87
£(9,11) 27.9219 28.0648 32983.47 14.8562 42.7781 8276.47 14.0705 42.1352 9124.66 50,384.61
£(8,12) 28.0419 28.1848 34,021.08 14.8162 42.8581 8206.85 14.0305 42.2152 9053.33 51,281.26
£(7,13) 28.1619 28.3048 35,059.98 14.7762 42.9381 8137.55 13.9905 42.2952 8982.31 52,179.83
RS 10 24.8444 24.9444 22,185.20 20.8111 45.6556 11,440.30 20.1111 45.0556 12,321.47 45,946.97
£(9,11) 24.9778 25.0778 23,308.72 20.7444 45.7222 11,342.67 20.0444 45.1222 12,221.84 46,873.23
£(8,12) 25.1111 25.2111 24,433.12 20.6778 45.7889 11,245.49 19.9778 45.1889 12,122.65 47,801.27
£(7,13) 25.2444 25.3444 25,558.42 20.6111 45.8556 11,148.75 19.9111 45.2556 12,023.91 48,731.08
• The wholesale prices will increase while the markup prices of the two retailers will drop when the uncertain degree of the manufacturing cost increases in the three structures.
• The manufacturer can benefit from the vagueness of the manufacturing costs, while the other channel members, namely the two retailers, will suffer from less profits.
• When the uncertain degree of the manufacturing cost increases, the whole supply chain will gain more, meanwhile, consumers have to pay more.
Referring to Tables 6 and 7, we obtain the following results:
• When the uncertain degree of its sales cost becomes higher, each retailer will be charged a lower wholesale price, while the wholesale price for the other retailer will keep the same in the MS structure and drop slightly in the RS and VN structures.
• Both the retailers can charge higher markup prices when the uncertain degree of either of the sales costs increases.
• Contrary to the effect of the manufacturing cost's uncertain degree, the retailers can gain higher profits while the common manufacturer will suffer from lower profit with the increase of the uncertain degree of the sales costs.
Table 6 Effects of the 1st retailer's sales cost's uncertain degree on the prices and profits
S1 w1 w2 nm n P1 n P2 n,2 nt
MS 6 32.3500 32.5667 34,351.97 13.3544 45.7044 5408.79 12.5511 45.1178 6596.68 46,357.43
£(5,7) 32.3167 32.5667 34,302.99 13.4056 45.7222 5956.74 12.5556 45.1222 6613.49 46,873.23
£(4,8) 32.2833 32.5667 34,254.12 13.4567 45.7400 6504.81 12.5600 45.1267 6630.32 47,389.24
£(3,9) 32.2500 32.5667 34,205.35 13.5078 45.7578 7052.98 12.5644 45.1311 6647.14 47,905.47
VN 6 27.9448 28.0686 33,030.55 14.8105 42.7552 7762.45 14.0629 42.1314 9096.45 49,889.45
£(5,7) 27.9219 28.0648 32,983.47 14.8562 42.7781 8276.47 14.0705 42.1352 9124.66 50,384.61
£(4,8) 27.8990 28.0610 32,936.47 14.9019 42.8010 8790.64 14.0781 42.1390 9152.89 50,880.00
£(3,9) 27.8762 28.0571 32,889.57 14.9476 42.8238 9304.94 14.0857 42.1429 9181.13 51,375.64
RS 6 24.9956 25.0822 23,341.49 20.7089 45.7044 10,817.57 20.0356 45.1178 12,198.37 46,357.43
£(5,7) 24.9778 25.0778 23,308.72 20.7444 45.7222 11,342.67 20.0444 45.1222 12,221.84 46,873.23
£(4,8) 24.9600 25.0733 23,275.99 20.7800 45.7400 11,867.93 20.0533 45.1267 12,245.32 47,389.24
£(3,9) 24.9422 25.0689 23,243.32 20.8156 45.7578 12,393.35 20.0622 45.1311 12,268.81 47,905.47
Table 7 Effects of the 2nd retailer's sales cost's uncertain degree on the prices and profits
S2_w W2 nm ri pi nn_r2 P2 nn_nt
MS 5 32.3167 32.6000 34,352.97 13.4011 45.7178 5950.10 12.5044 45.1044 6091.67 46,394.74
£(4.5,5.5) 32.3167 32.5833 34,327.97 13.4033 45.7200 5953.42 12.5300 45.1133 6352.58 46,633.97
£(4,6) 32.3167 32.5667 34,302.99 13.4056 45.7222 5956.74 12.5556 45.1222 6613.49 46,873.23
£(3.5,6.5) 32.3167 32.5500 34,278.04 13.4078 45.7244 5960.07 12.5811 45.1311 6874.41 47,112.52
VN 5 27.9257 28.0876 33,031.37 14.8486 42.7743 8262.93 14.0248 42.1124 8641.06 49,935.37
£(4.5,5.5) 27.9238 28.0762 33,007.41 14.8524 42.7762 8269.70 14.0476 42.1238 8882.86 50,159.97
£(4,6) 27.9219 28.0648 32,983.47 14.8562 42.7781 8276.47 14.0705 42.1352 9124.66 50,384.61
£(3.5,6.5) 27.9200 28.0533 32,959.55 14.8600 42.7800 8283.25 14.0933 42.1467 9366.47 50,609.27
RS 5 24.9822 25.0956 23,341.89 20.7356 45.7178 11,329.51 20.0089 45.1044 11,723.34 46,394.74
£(4.5,5.5) 24.9800 25.0867 23,325.30 20.7400 45.7200 11,336.09 20.0267 45.1133 11,972.58 46,633.97
£(4,6) 24.9778 25.0778 23,308.72 20.7444 45.7222 11,342.67 20.0444 45.1222 12,221.84 46,873.23
£(3.5,6.5) 24.9756 25.0689 23,292.15 20.7489 45.7244 11,349.26 20.0622 45.1311 12,471.12 47,112.52
• In the same way, when the uncertain degree of the sales costs increases, end consumers will afford higher prices while the whole supply chain will gain more.
Similarly, more experiments can be conducted to explore the effects of the other parameters, such as the two price elastic coefficients or the market sizes of the two retailers.
Conclusions
In this paper, we considered a pricing competing problem in supply chain with competing retailers. Specially, the manufacturing costs, sales costs, and consumer demands were characterized as uncertain variables. Meanwhile, three decentralized models based on uncertainty theory and game theory were built to formulate the pricing decision problems. The equilibrium behaviors of the supply chain under three possible power structures were derived from these models. Afterwards, numerical experiments were also given to explore the impacts of uncertain degrees of the parameters on the pricing decisions. The results showed that the existence of dominant power in a supply chain will increase the sales prices and lower the profits of the whole supply chain. It was also found that the supply chain members may benefit from higher uncertain degrees of their own costs, whereas the supply chain members at the other level will gain less profits. Additionally, the results demonstrated that the uncertainty of the supply chain will make end consumers pay more.
As this study was based on some assumptions, future researches can focus on some more general problems. For instance, this paper only considered one type of indeterminacy, while the real world might behave more complicated in which randomness and uncertainty might coexist. Therefore, one possible extension of this paper is to study the pricing problem with twofold indeterminacy, in which uncertain random variable can be applied. Besides, this paper assumed that all the participants are risk neutral while the decision makers may be risk sensitive in the real world. The research can be more applicable if the equilibrium behaviors with risk-sensitive members are considered.
Acknowledgements
This work was supported by National Natural Science Foundation of China (No. 71371141) and the Fundamental Research Funds for the Central Universities.
Authors' Contributions
YW and HK carried out the study in the paper and drafted the first version of the manuscript. HH designed the framework of the paper and with ZC carried out the numerical experiments. All authors read and approved the final manuscript.
Competing Interests
The authors declare that they have no competing interests.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Author details
1 School of Economics and Management, Tongji University, Siping Road, Shanghai 200092, China. 2Department of Geotechnical Engineering, Tongji University, Siping Road, Shanghai 200092, China.
Received: 5 February 2017 Accepted: 5 March 2017 Published online: 20 March 2017
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